Permanence and extinction of periodic predator–prey systems in a patchy environment with delay

This paper studies two species predator–prey Lotka–Volterra type dispersal systems with periodic coefficients and infinite delays, in which the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. Sufficient and necessary conditions of integrable form for the permanence, extinction and the existence of positive periodic solutions are established, respectively. Some well-known results on the nondelayed periodic predator–prey Lotka–Volterra type dispersal systems are improved and extended to the delayed case.     

An impulsive periodic predator–prey system with Holling type III functional response and diffusion

This paper studies an impulsive two species periodic predator–prey Lotka–Volterra type dispersal system with Holling type III functional response in a patchy environment, in which the prey species can disperse among n different patches, but the predator species is confined to one patch and cannot disperse. Conditions for the permanence and extinction of the predator–prey system, and for the existence of a unique globally stable periodic solution are established. Numerical examples are shown to verify the validity of our results.     

Permanence for a class of periodic time-dependent predator–prey system with dispersal in a patchy-environment

In this paper, we study two species predator–prey Lotka–Volterra type dispersal system with periodic coefficients in two patches, in which both the prey and predator species can disperse between two patches. By utilizing analytic method, sufficient and realistic conditions on permanence and the existence of periodic solution are established. The theoretical results are confirmed by a special example and numerical simulations.     

PERMANENCE OF A NONLINEAR PERIODIC PREDATOR-PREY SYSTEM WITH PREY DISPERSAL AND PREDATOR DENSITY-INDEPENDENCE

In this paper,a set of suffcient conditions which ensure the permanence of a nonlinear periodic predator-prey system with prey dispersal and predator density-independence are obtained,where the prey species can disperse among n patches,while the density-independent predator is confined to one of the patches and cannot disperse. Our results generalize some known results.     

Persistence in nonautonomous predator–prey systems with infinite delays

This paper studies the general nonautonomous predator–prey Lotka–Volterra systems with infinite delays. The sufficient and necessary conditions of integrable form on the permanence and persistence of species are established. A very interesting and important property of two-species predator–prey systems is discovered, that is, the permanence of species and the existence of a persistent solution are each other equivalent. Particularly, for the periodic system with delays, applying these results, the sufficient and necessary conditions on the permanence and the existence of positive periodic solutions are obtained. Some well-known results on the nondelayed periodic predator–prey Lotka–Volterra systems are strongly improved and extended to the delayed case.     

Permanence and periodicity of a delayed ratio-dependent predator-prey model with stage structure

A periodic ratio-dependent predator-prey model with time delays and stage structure for both prey and predator is investigated. It is assumed that immature individuals and mature individuals of each species are divided by a fixed age, and that immature predators do not have the ability to attack prey. Sufficient conditions are derived for the permanence and existence of positive periodic solutions of the model. Numerical simulations are presented to illustrate the feasibility of our main results.     

Permanence and extinction analysis for a delayed periodic predator-prey system with Holling type II response function and diffusion

In this paper, it is studied that two species predator-prey Lotka-Volterra type dispersal system with delay and Holling type II response function, in which the prey species can disperse among n patches, while the density-independent predator species is confined to one of the patches and cannot disperse. Sufficient conditions of integrable form for the boundedness, permanence, extinction and the existence of positive periodic solution are established, respectively.     

Permanence for a delayed periodic predator-prey model with prey dispersal in multi-patches and predator density-independent

In this paper, we study two species time-delayed predator-prey Lotka-Volterra type dispersal systems with periodic coefficients, in which the prey species can disperse among n patches, while the density-independent predator species is confined to one of patches and cannot disperse. Sufficient conditions on the boundedness, permanence and existence of positive periodic solution for this systems are established. The theoretical results are confirmed by a special example and numerical simulations.     

Permanence and extinction for dispersal population systems

This paper studies the effect of dispersal on the permanence of population models in poor patchy environment. We first consider the logistic system with dispersal for single species and obtain the conditions for its permanence. On the basis of the conditions, we then consider a periodic predator-prey system where the prey can disperse among several patches. A necessary and sufficient condition is obtained for the permanence of the periodic predator-prey system. We discuss the biological implications of the main results.     

A study of predator–prey models with the Beddington–DeAnglis functional response and impulsive effect

In this paper, the predator–prey system with the Beddington–DeAngelis functional response is developed, by introducing a proportional periodic impulsive catching or poisoning for the prey populations and a constant periodic releasing for the predator. The Beddington–DeAngelis functional response is similar to the Holling type II functional response but contains an extra term describing mutual interference by predators. This model has the potential to protect predator from extinction, but under some conditions may also lead to extinction of the prey. That is, the system exists a locally stable prey-eradication periodic solution when the impulsive period satisfies an inequality. The condition for permanence is established via the method of comparison involving multiple Liapunov̀ functions. Further, by numerical simulation method the influences of the impulsive perturbations and mutual interference by predators on the inherent oscillation are investigated. With the increasing of releasing for the predator, the system appears a series of complex phenomenon, which include (1) period-doubling, (2) chaos attractor, (3) period-halfing. (4) non-unique dynamics (meaning that several attractors coexist).     

Permanence,extinction and periodic solution of the predator–prey system with Beddington–DeAngelis functional response and stage structure for prey

In this paper, we study the permanence, extinction and periodic solution of the periodic predator–prey system with Beddington–DeAngelis functional response and stage structure for prey. A set of sufficient and necessary conditions which guarantee the predator and prey species to be permanent are obtained. In addition, sufficient conditions are derived for the existence of positive periodic solutions to the system. Numeric simulations show the feasibility of the main results.     

Permanence and periodicity of a delayed ratio-dependent predator–prey model with Holling type functional response and stage structure

A periodic and delayed ratio-dependent predator–prey system with Holling type III functional response and stage structure for both prey and predator is investigated. It is assumed that immature predator and mature individuals of each species are divided by a fixed age, and immature predator do not have the ability to attack prey. Sufficient conditions are derived for the permanence and existence of positive periodic solution of the model. Numerical simulations are presented to illustrate the feasibility of our main results.     

Existence of periodic solutions in predator-prey with Watt-type functional response and impulsive effects

In this paper, the existence of a periodic solution and the permanence of a periodic Watt-type predator-prey system where the prey disperses in patchy environment with two patches are investigated. With the continuation theorem of coincidence degree theory, we obtain the existence of a positive periodic solution for such a system. We assume the Watt-type functional response within-patch dynamics and provide a sufficient condition to guarantee the predator and prey species to be permanent. Furthermore, we give numerical analysis to confirm our theoretical results. It will be useful to ecosystem control.     

The dynamical behavior of a predator–prey system with Gompertz growth function and impulsive dispersal of prey between two patches

In this paper, we investigate a predator–prey model with Gompertz growth function and impulsive dispersal of prey between two patches. Using the dynamical properties of single‐species model with impulsive dispersal in two patches and comparison principle of impulsive differential equations, necessary and sufficient criteria on global attractivity of predator‐extinction periodic solution and permanence are established. Finally, a numerical example is given to illustrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

On Nonautonomous Prey predator Patchy System

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Permanence, extinction and periodic solution of predator-prey system with Beddington-DeAngelis functional response

In this paper we study permanence, extinction and periodic solution of periodic predator-prey system with Beddington-DeAngelis functional response. We provide a sufficient and necessary condition to guarantee the predator and prey species to be permanent. In addition, sufficient condition is derived for the existence of positive periodic solution. This paper improves some main results obtained by Fan and Kuang [M. Fan, Y. Kuang, Dynamics of nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl. 295 (2004) 15-39].     

Periodic solutions of a nonautonomous predator–prey system with stage structure and time delays

A nonautonomous Lotka–Volterra type predator–prey model with stage structure and time delays is investigated. It is assumed in the model that the individuals in each species may belong to one of two classes: the immatures and the matures, the age to maturity is presented by a time delay, and that the immature predators do not feed on prey and do not have the ability to reproduce. By some comparison arguments we first discuss the permanence of the model. By using the continuation theorem of coincidence degree theory, sufficient conditions are derived for the existence of positive periodic solutions to the model. By means of a suitable Lyapunov functional, sufficient conditions are obtained for the uniqueness and global stability of the positive periodic solutions to the model.     

A predator–prey system with a stage structure for the prey

This paper considers a periodic predator–prey system where the prey has a life history that takes the prey through two stages: immature and mature. We provide a sufficient and necessary condition to guarantee permanence of the system. It is shown that the system is permanent if and only if the growth of the predator by foraging the prey minus its death rate is positive on average during the period.     

Stability and bifurcation of a stage-structured predator–prey model with both discrete and distributed delays

This paper concerns with a new delayed predator–prey model with stage structure on prey, in which the immature prey and the mature prey are preyed by predator and the delay is the length of the immature stage. Mathematical analysis of the model equations is given with regard to invariance of non-negativity, boundedness of solutions, permanence and global stability and nature of equilibria. Our work shows that the stage structure on the prey is one of the important factors that affect the extinction of the predator, and the predation on immature prey is a cause of periodic oscillation of population and can make the behaviors of the system more complex. The predation on the immature and mature prey brings both positive and negative effects on the permanence of the predator, if ignore the predation on immature prey in the system, the stage-structure on prey brings only negative effect on the permanence of the predator.     

Uniqueness of periodic solutions of a nonautonomous density-dependent predator–prey system

In this present paper, we investigate the uniqueness of periodic solutions of a nonautonomous density-dependent and ratio-dependent predator–prey system, where not only the prey density dependence but also the predator density dependence are considered, such that the studied predator–prey system conforms to the realistically biological environment. We start with a sufficient condition for the permanence of the system and then construct a weaker sufficient condition by introducing a specific set, denoted as Γ. Based on this Γ and the Brouwer fixed-point theorem, we obtain the existence condition of positive periodic solutions. Moreover, since the uniqueness of positive periodic solutions can be ensured by global attractiveness, we alternatively introduce a sufficient condition for global attractiveness. Similarly, we also provide a sufficient condition for the uniqueness of non-negative periodic solutions.